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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Characterization of scaling functions in a multiresolution analysis
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by P. Cifuentes, K. S. Kazarian and A. San Antolín PDF
Proc. Amer. Math. Soc. 133 (2005), 1013-1023 Request permission

Abstract:

We characterize the scaling functions of a multiresolution analysis in a general context, where instead of the dyadic dilation one considers the dilation given by a fixed linear map $A: \mathbb {R}^n\rightarrow \mathbb {R}^n$ such that $A(\mathbb {Z}^n) \subset \mathbb {Z}^n$ and all (complex) eigenvalues of $A$ have absolute value greater than $1.$ In the general case the conditions depend on the map $A.$ We identify some maps for which the obtained condition is equivalent to the dyadic case, i.e., when $A$ is a diagonal matrix with all numbers in the diagonal equal to $2.$ There are also easy examples of expanding maps for which the obtained condition is not compatible with the dyadic case. The complete characterization of the maps for which the obtained conditions are equivalent is out of the scope of the present note.
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Additional Information
  • P. Cifuentes
  • Affiliation: Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain
  • Email: patricio.cifuentes@uam.es
  • K. S. Kazarian
  • Affiliation: Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain
  • Email: kazaros.kazarian@uam.es
  • A. San Antolín
  • Affiliation: Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain
  • Email: angel.sanantolin@uam.es
  • Received by editor(s): March 6, 2003
  • Received by editor(s) in revised form: June 19, 2003
  • Published electronically: November 19, 2004
  • Additional Notes: The first two authors were partially supported by BFM2001-0189
  • Communicated by: David R. Larson
  • © Copyright 2004 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 133 (2005), 1013-1023
  • MSC (2000): Primary 42C15; Secondary 42C30
  • DOI: https://doi.org/10.1090/S0002-9939-04-07786-X
  • MathSciNet review: 2117202